Aliquot sequences

Aliquot sequences arise in iterating the sum-of-divisors function s,
which assigns to a positive integer the sum of its proper divisors
(i.e., excluding the number itself). An aliquot sequence thus starts with
a positive integer n, followed by s(n), then s(s(n)), etc.

The main open question in the area is given by the Catalan-Dickson
conjecture, which states that for any starting value n one of two
things happens: either the sequence ends in a cycle, or it reaches 1
after finitely many steps (and is said to terminate then).

Progress in numerical work to verify the Catalan-Dickson conjecture
relies upon progress in integer factorization methods, as no better
method is known for the determination of the divisors of a number than
the obvious method using its prime factorization.

I became interested in aliquot sequences while testing the integer
factorization algorithms in Magma. I have listed some of the
numerical work
I have done. Several other people are involved in a big effort
to push our knowledge for small starting values.

Recent work on aliquot sequences

Older work on aliquot sequences

For all n up to 50000 I have computed terms of the aliquot
sequence starting with n until it terminated, entered an aliquot
cycle, or reached an entry with at least 80 decimal digits. The aim
now is to continue until at least 90 digits (or termination) is reached.
All this is ongoing work as part of a bigger effort by several people
(see the aliquot page of Wolfgang Creyaufmüller, or the page of Juan Varona, or the
aliquot page
of Paul Zimmermann).

Sequences with starting values up to 50000

Here is a summary of the status of my own computations.

Not(-yet)-terminating aliquot sequences with starting values up to 10000,
summarized in
table 1.
This includes the current status of each of these sequences, as established
by the work of Juan Varona and Manuel Benito,
see their
page, and that of Paul Zimmermann, who holds the current records for
the five starting values less than 1000 remaining (the `Lehmer five')
for which it is not yet known whether the sequence terminates, as well as for
the sequences starting with 1074 and 1134; see his
aliquot page.